Matrix generalizations of integrable systems with Lax - - PowerPoint PPT Presentation

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Matrix generalizations of integrable systems with Lax integro-differential representations

Chvartatskyi O.1 advisor Yu. M. Sydorenko1

1IVAN FRANKO NATIONAL UNIVERSITY OF LVIV

XIV-th International Conference Geometry, Integrability and Quantization Varna, Bulgaria, 8–13 June, 2012

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Contents

1 Introduction 2 Symmetry reductions of the KP–hierarchy 3 Exact solutions of some nonlinear models from the

KP-hierarchy

4 Integro-differential Lax representations for Davey-Stewartson

and Chen-Lee-Liu equations

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Introduction We consider linear space ζ of micro-differential operators over the field С of the following form: L ∈ ζ =   

n(L)

• i=−∞

aiDi : n(L) ∈ Z    , (1) where coefficients ai are functions dependent on spatial variable x = t1 and evolution parameters t2, t3 . . . . Coefficients ai(t), t = (t1, t2, . . .), are supposed to be smooth functions of vector variable t that has a finite number of elements that belong to some functional space A. This space is a differential algebra under arithmetic operations. An operator of differentiation is denoted in the following way: D :=

∂ ∂x .

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Introduction Addition and multiplication of operators by scalars (elements of the field C) are introduced in the following way: λ1L1 ± λ2L2 =

N1

• i=−∞

λ1a1iDi ±

N2

• i=−∞

λ2a2iDi =

max<N1,N2>

• i=−∞

(λ1a1i ± λ2a2i)Di, λ1, λ2 ∈ C. The structure of Lie algebra on a linear space ζ (1) is defined by the commutator [·, ·] : ζ × ζ → ζ , [L1, L2] = L1L2 − L2L1, where the composition of micro-differential operators L1 and L2 is induced by general Leibniz rule:

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Introduction Dnf :=

∞

• j=0

n j

• f (j)Dn−j,

(2) n ∈ Z, f ∈ A ⊂ ζ, f (j) := ∂jf

∂xj ∈ A ⊂ ζ, DnDm = DmDn = Dn+m,

n, m ∈ Z, where n

• := 1,
• n

j

• := n(n−1)...(n−j+1)

j!

. Formula (2) defines the composition of the operator Dn ∈ ζ and the operator of multiplication by function f ∈ A ⊂ ζ in contradistinction to the denotation Dk{f} := ∂kf

∂xk ∈ A, k ∈ Z+.

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Introduction Consider a microdifferential Lax operator: L := WDW −1 = D +

∞

• i=1

UiD−i, (3) which is parametrized by the infinite number of dynamic variablesUi = Ui(t1, t2, t3, ...), i ∈ N, which depend on an arbitrary (finite) number of independent variables t1 := x, t2, t3, ... All dynamic variables Ui can be expressed in terms of functional coefficients of formal dressing Zakharov-Shabat

• perator:

W = I +

∞

• i=1

wiD−i, (4) The inverse of formal operator W has the form: W −1 = I +

∞

• i=1

aiD−i. (5)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Introduction In scalar case, Kadomtsev -Petviashvili hierarchy is a commuting family of evolution Lax equations for the operator L (3) αiLti = [Bi, L] := BiL − LBi, (6) where αi ∈ C, i ∈ N, the operator Bi := (Li)+ is a differential part of the i-th power of microdifferential symbol L. By symbol Lti we will denote the following operator: Lti := (WDW −1)ti =

∞

• j=1

(Uj)tiD−j. (7) Formally transposed and conjugated operators Lτ, L∗ have the form:

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Introduction Lτ := −D +

∞

• j=1

(−1)jD−jUj, L∗ := ¯ Lτ. (8) Zakharov-Shabat equations are consequences of the commutativity of two arbitrary flows in (6) with i = m and i = n Ltmtn = Ltntm ⇒ ⇒ [αn∂tn−Bn, αm∂tm−Bm] = αmBntm−αnBmtn+[Bn, Bm] = 0. (9)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Symmetry reductions of the KP–hierarchy References Sidorenko Yu., Strampp W. Symmetry constraints of the KP–hierarchy // Inverse Problems. – 1991. – V. 7. –

• P. L37-L43.

Konopelchenko B., Sidorenko Yu., Strampp W. (1+1)-dimensional integrable systems as symmetry constraints of (2+1)-dimensional systems // Phys. Lett. A. – 1991. – V. 157. – P. 17-21. Cheng Yi. Constrained of the Kadomtsev – Petviashvili hierarchy // J. Math. Phys. – 1992. – Vol.33. –

• P. 3774-3787.

Sidorenko Yu., Strampp W. Multicomponent integrable reductions in Kadomtsev-Petviashvilli hierarchy // J.

• Math. Phys. – 1993. – V. 34, №4. – P. 1429-1446.

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Symmetry reductions of the KP–hierarchy References

• A. M. Samoilenko, V. G. Samoilenko and Yu. M. Sidorenko

Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: Many-dimensional generalizations and exact solutions of reduced system // Ukr. Math. Journ., 1999, Vol. 51, № 1, p. 86-106

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Symmetry reductions of the KP–hierarchy Consider a symmetry reduction of the KP-hierarchy, which is a generalization of the Gelfand-Dickey k-reduction: (Lk)− := (Lk)<0 = qM0D−1r⊤ = = x q(x, t2, t3, . . .)M0r⊤(s, t2, t3, . . .) · ds, (10) where Matl×l(C) ∋ M0 is a constant matrix, and functions q = (q1, ..., ql), r = (r1, ..., rl) are fixed solutions of the following system of differential equations: αnqtn = Bn{q}, αnrtn = −Bτ

n{r},

(11) where n ∈ N.

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Symmetry reductions of the KP–hierarchy Reduced flows (6), (10), (11) admit Lax representation [Lk, Mn] = 0, Lk = Bk + qM0D−1r⊤, Mn = αn∂tn − Bn. (12) Equation (12) is equivalent to the (1 + 1)-dimensional integrable systems for functional coefficients Ui , i = 1, k − 1 and functions q, r: Uitn = Pin[U1, U2, ..., Uk−1, q, r], qtn = Bn[Ui, q, r]{q}, rtn = −Bτ

n[Ui, q, r]{r},

(13) where i = 1, k − 1, Pin and Bn are differential polynomials with respect to dynamic variables that are indicated in square brackets.

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Symmetry reductions of the KP–hierarchy (2+1)-dimensional generalizations of Lax representations (12) have the form: [Lk, Mn] = 0, (14) where Lk is (2+1)-dimensional integro-differential operator: Lk = α∂y − Bk − qM0D−1r⊤, (15) and Mn in (14) is evolutional differential operator of n-th order with respect to spatial variable x: Mn = αn∂tn −

n

• j=1

vjDj (16)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Symmetry reductions of the KP–hierarchy Consider examples of equations (12)-(13) and their generalizations (14)-(16) for some k and n:

• 1. k = 1, n = 2 :

L1 = D + qM0D−1q∗,M2 = α2∂t2 − D2 − 2qM0q∗, where α2 ∈ iR, M∗

0 = M0.

Equation [L1, M2] = 0 is equivalent to nonlinear Schrodinger equation (NLS): α2qt2 = qxx + 2 (qM0q∗) q. (17)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Symmetry reductions of the KP–hierarchy Now let us consider spatially two-dimesional generalizations of the operators L1, M2: L1 = ∂y − qM0D−1q∗, M2 = α2∂t2 − c1D2 − 2c1S1, (18) where α2 ∈ iR, S1 = S1(x, y, t2) = ¯ S1(x, y, t2),c1 ∈ R Lax equation [L1, M2] = 0 is equivalent to Davey-Stewartson system DS-III: α2qt2 = c1qxx − 2c1S1q S1y = (qM0q∗)x (19) System (19) is spatially two-dimensional l–component generalization of NLS.

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Symmetry reductions of the KP–hierarchy

• 2. k = 2, n = 2 :

L2 = D2 + 2u + qM0D−1q∗, M2 = α2∂t2 − D2 − 2u, where M∗

0 = −M0, u = ¯

u, α2 ∈ iR. Operator equation [L2, M2] = 0 is equivalent to Yajima-Oikawa system: α2qt2 = qxx + 2uq, α2ut2 = (qM0q∗)x. (20)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Symmetry reductions of the KP–hierarchy (2+1)-dimensional generalization of the operators L2, M2 have a form: L2 = i∂y − D2 − 2u − qM0D−1q∗, M2 = α2∂t2 − D2 − 2u, where α2 ∈ iR, M0 = −M∗

0, u = ¯

u. Equation [L2, M2] = 0 can be represented in the following way: α2ut2 = iuy + (qM0q∗)x α2qt2 = qxx + 2uq. (21) System (21) is l–component spatially two-dimensional generalization of the Yajima-Oikawa system.

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Symmetry reductions of the KP–hierarchy

• 3. k = 2, n = 3 :

L2 = D2 + 2u + qM0D−1q∗, M3 = α3∂t3 − D3 − 3uD − 3

2(ux + qM0q∗),

where M0 = −M∗

0, u = ¯

u, α3 ∈ R. Equation [L2, M3] = 0 is equivalent to the system: α3qt3 = qxxx + 3uqx + 3

2uxq + 3 2qM0q∗q,

α3ut3 = 1

4uxxx + 3uux + 3 4 (qxM0q∗ − qM0q∗ x)x .

(22)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Symmetry reductions of the KP–hierarchy Spatially two-dimensional generalizations of L2 and M3 have the form: L2 = i∂y − D2 − 2u − qM0D−1q∗, M3 = α3∂t3 − D3 − 3uD − 3

2

• ux + iD−1{uy} + qM0q∗

, (23) wher α3 ∈ R, M∗

0 = −M0, u = ¯

• u. Equation [L2, M3] = 0 is

equivalent to the system:      α3qt3 = qxxx + 3uqx + 3

2

• ux + iD−1{uy} + qM0q∗

q,

• α3ut3 − 1

4uxxx − 3uux + 3 4 (qM0q∗ x − qxM0q∗)x +

− 3

4i (qM0q∗)y

• x = − 3

4uyy.

(24) Equations (24) generalize Mel’nikov system.

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Symmetry reductions of the KP–hierarchy References V.K.Mel’nikov. On equations for wave interactions. Lett.Math.Phys. 7:2 (1983) 129–136.

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Symmetry reductions of the KP–hierarchy In other cases we will obtain:

• 4. Vector generalization of the modified Korteweg-de Vries

equation (k = 1, n = 3): α3qt3 = qxxx +3 (qM0q∗) qx +3 (qxM0q∗) q, M∗

0 = M0. (25)

• 5. Generalization of the Boussinesq equation (k = 3, n = 2):

3α2

2ut2t2 = (−uxx − 6u2 + 4(qM0q∗))xx, M∗ 0 = M0,

α2qt2 − qxx − 2uq = 0, (26)

• 6. Vector generalization of the Drinfeld-Sokolov system

(k = 3, n = 3):

• α3qt3 = qxxx + 3uqx + 3

2uxq, q = ¯

q, M∗

0 = M0,

α3ut3 = (qM0q⊤)x, (27)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Exact solutions of some nonlinear models from the KP-hierarchy In this section we will consider the construction of exact solutions of the integrable systems from the KP-hierarchy. For this reason we will use invariant transformations for linear integro-differential operators from the previous section. Consider the integro-differential operator: L := α∂t −

n

• i=0

uiDi + qM0D−1r⊤, α ∈ iR ∪ R (28) with (N × N)-matrix coefficients ui = ui(x, t); Λ, ˜ Λ and M0 are (K × K) and (l × l)-matrices correspondingly; q, r are (N × l)-matrices. Assume that (N × K)-matrix functions ϕ, ψ satisfy linear problems: L{ϕ} = ϕΛ, Lτ{ψ} = ψ˜ Λ.

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Define the binary Darboux-type transformation (BT) as: W = I − ϕ

• C + D−1{ψ⊤ϕ}

−1 D−1ψ⊤ (29) The following theorem holds: Theorem 1 Let functions f and g be (N × 1)-solutions of the linear systems L{f} = fλ, Lτ{g} = g˜ λ. Then, functions F = W{f}, G = W −1,τ{g} satisfy equations ˆ L{F} = Fλ, ˆ L−1,τ{G} = G˜ λ with the operator ˆ L = α∂t −

n

• i=0

ˆ uiDi + ˆ qM0D−1ˆ r⊤ + ΦMD−1Ψ⊤, (30) where M = CΛ − ˜ Λ⊤C, Φ = ϕ

• C + D−1{ψ⊤ϕ}

−1, Ψ⊤ =

• C + D−1{ψ⊤ϕ}

−1 ψ⊤, ˆ q = W{q}, ˆ r = W −1,τ{r}.

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Exact solutions of some nonlinear models from the KP-hierarchy References Sydorenko Yu. Generalized Binary Darboux–like Theorem for Constrained Kadomtsev–Petviashvili (cKP) Flows // Proceedings of Institute of Mathematics of NAS of Ukraine. – 2004. – V. 50, Part 1. – P. 470-477. Sydorenko Yu., Chvartatskyi O. Binary transformations for spatially two-dimensional operators and Lax equations //

• Visn. Kyiv. Univ. Ser: mech.-mat. - 2009. – V . 22. p. 32-35

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Exact solutions of some nonlinear models from the KP-hierarchy Consider two possible realizations of the integration operator D−1 in BT (29): W + = I − ϕ

• C +

x

−∞

ψ⊤(s)ϕ(s)ds −1 x

−∞

ψ⊤(s) · ds, (31) W − = I − ϕ

• C +

x

−∞

ψ⊤(s)ϕ(s)ds −1 x

+∞

ψ⊤(s) · ds, (32) under assumption that the components of (N × K)-matrix functions ϕ and ψ decrease rapidly at both infinities. A composition of operators (W +)−1 and W − gives Fredholm

• perator:

SR = (W +)−1W − = I + ϕC−1 +∞

−∞

ψ⊤(s) · ds. (33)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Exact solutions of some nonlinear models from the KP-hierarchy Assume that integral part in L is equal to zero. Using the equalities L{ϕ} = ϕΛ, Lτ{ψ} = ψ˜ Λ we obtain that the commutator of SR and L has the form: [L, SR] = ϕC−1M +∞

−∞

C−1ψ⊤(s) · ds, M = CΛ − ˜ Λ⊤C Using W +, W − as the dressing operators for L we obtain that: ˆ L1 = W +L(W +)−1 = (ˆ L1)+ + ΦM x

−∞ Ψ⊤(s) · ds,

ˆ L2 = W −L(W −)−1 = (ˆ L2)+ + ΦM x

+∞ Ψ⊤(s) · ds,

(ˆ L1)+ = (ˆ L2)+ (34)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Exact solutions of some nonlinear models from the KP-hierarchy If we put Λ = ˜ Λ = 0 and consider the differential operator L (M0 = 0), then using transformations W + or W − we obtain the differential operator ˆ

• L. In this case, [L, SR] = 0. Thus, we
• btain dressing due to Zakharov-Shabat.

References V.E. Zakharov, A.B. Shabat. A scheme for integrating the nonlinear equations of mathematical physics by the method

• f the inverse scattering problem. I, Funct. Anal. Appl.,

8(3), 226-235, 1974.

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Exact solutions of some nonlinear models from the KP-hierarchy Now we will consider realizations of integral transformation W (29) and construction of the solutions for integrable systems from the KP-hierarchy. At first we will consider the scalar NLS iqt = qxx + 2µ|q|2q, µ = ±1, (35) and its vector generalization – Manakov system (l components): i(qj)t = (qj)xx + 2

• l
• s=1

µs|q|2

s

• qj, µs = ±1, j = 1, l.

(36)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Exact solutions of some nonlinear models from the KP-hierarchy References S.V. Manakov. On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Sov. Phys. JETP 38:2 (1974) 248–253.

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Exact solutions of some nonlinear models from the KP-hierarchy Proposition 1 Let function ϕ := (ϕ1, . . . , ϕK) be a fixed solution of the system: ϕx = ϕΛ, iϕt = ϕxx, (37) where Λ ∈ MatK×K(C). Let f := (f1, . . . , fl) be an arbitrary solution of the problem ift = fxx. (38) Then functions F := f − ϕ(C + Ω[ ¯ ϕ, ϕ])−1Ω[ ¯ ϕ, f], Φ = ϕ(C + Ω[ ¯ ϕ, ϕ])−1, where Ω[ ¯ ϕ, ϕ] = (x,t)

(x0,t0) ϕ∗ϕdx + i(ϕ∗ xϕ − ϕ∗ϕx)dt,

Ω[ ¯ ϕ, f] = (x,t)

(x0,t0) ϕ∗fdx + i(ϕ∗ xf − ϕ∗fx)dt, C = C∗ ∈ MatK×K(C)

satisfy equations:

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Exact solutions of some nonlinear models from the KP-hierarchy iFt = Fxx + 2Φ ˆ MΦ∗F, (39) iΦt = Φxx + 2Φ ˆ MΦ∗Φ, (40) where ˆ M = CΛ + Λ∗C − (ϕ∗ϕ)(x0, t0)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Exact solutions of some nonlinear models from the KP-hierarchy Using proposition 1, we can obtain K-soliton solution of NLS (µ = 1) of the following structure: q = det

• ∆2

− → 1 ϕ

• det(∆2)

, where ϕj = γjeλjx+iλ2

j t, γj, λj ∈ C, j = 1, K; −

→ 1 is a row-vector (K-components) consisting of 1, ∆2 =

• 1

λs + ¯ λj ( ¯ ϕjϕs + 1) K

j,s=1

Animation 1 describes the behavior of 3-soliton solution (|q| and Re(q) ) with λ1 = 1.5 + i, λ2 = 1 + 2i, λ3 = 2.5 + 3.5i and γ1 = e, γ2 = e10, γ3 = e5.

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Exact solutions of some nonlinear models from the KP-hierarchy We can also use Proposition 1 for obtaining other kinds of solutions (e.g. bound states) for NLS and constructing solutions

• f vector generalization of NLS.

Animation 2 describes the behavior of NLS solution, consisting

• f 1 bound state and 1 soliton.

Animation 3 represents the absolute value of the solution (λ1 = 2 − 3i, λ2 = 1 + 2i, γ1 = e100, γ2 = e10) for 2-component NLS generalization of the form: i(qj)t = (qj)xx + 2

• |q1|2 − |q2|2

qj, j = 1, 2 (41)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Exact solutions of some nonlinear models from the KP-hierarchy Similar types of solutions for other integrable systems of the KP-hierarchy can also be constructed. In particular, one of bound-state solutions of the Yajima-Oikawa system iqt2 = qxx + 2uq. iut2 = (µ|q|2)x; (42) in case µ = −i is presented on animation 4 (λ = 3 + i, γ = e5). Animation 5 describes the behavior of 2-soliton solution of Drinfeld-Sokolov system:

• qt3 = qxxx + 3uqx + 3

2uxq,

ut3 = (q2)x. (43)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Exact solutions of some nonlinear models from the KP-hierarchy References Sydorenko Yu., Chvartatskyi O. Integration of scalar Kadomtsev-Petviashvili kierarchy by the method of integral Darboux-like transformations // Visn. Lviv. Univ. Ser: mech.-mat. - 2011. – V . 75. p. 181-225

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Integro-differential Lax representations for Davey-Stewartson and Chen-Lee-Liu equations Consider the generalizations of operators L1, M2 (18): L1 = ∂y − qM0D−1q∗, M2 = α2∂t2 − c1D2 − c2∂2

y + 2c1S1 + 2c2qM0D−1∂yq∗,

(44) where c1, c2 ∈ R, α2 ∈ iR, q = q(x, y, t) and S1 = S1(x, y, t) = S∗

1(x, y, t) are matrix functions with

dimensions N × l and N × N respectively; M0 = M∗

0 is a

constant (l × l)-dimensional matrix. Lax equation [L1, M2] = 0 is equivalent to the system: α2qt2 = c1qxx + c2qyy − 2c1S1q − 2c2qM0S2, S1y = (qM0q∗)x, S2x = (q∗q)y. (45)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Integro-differential Lax representations for Davey-Stewartson and Chen-Lee-Liu equations In scalar case (N = 1, l = 1), by taking S = c1S1 + c2S2, µ := M0 = 1, we obtain the following differential consequence from (45): α2qt2 = c1qxx + c2qyy − 2Sq, Sxy = c1|q|2

xx + c2|q|2 yy.

(46) If c1 = −c2 = c ∈ R we obtain Davey-Stewartson system (DS-I) from (46).

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Integro-differential Lax representations for Davey-Stewartson and Chen-Lee-Liu equations Consider the following pair of operators: L1 = ∂¯

z − qD−1 z ¯

q, M2 = α2∂t2−cD2

zz+c∂2 ¯ z¯ z+2cS1−2cqD−1 z ¯

q¯

z−2cqD−1 z ¯

q∂¯

z, (47)

where α2, c ∈ iR; q and S1 are (N × N)-matrices, z = x + iy. Lax equation [L1, M2] = 0 is equivalent to the system:

• α2qt2 = −icqxy − 2cS1q + 2cq¯

S1, S1x + iS1y = (q¯ q)x − i(q¯ q)y. (48)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Integro-differential Lax representations for Davey-Stewartson and Chen-Lee-Liu equations In scalar case (N = 1) we obtain the following differential consequence from system (48):

• α2qt2 = −icqxy − 4ic ˜

Sq, ˜ Sxx + ˜ Syy = −4|q|2

xy.

(49) System (49) is Davey-Stewartson system (DS-II).

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Integro-differential Lax representations for Davey-Stewartson and Chen-Lee-Liu equations References Athorne C., Fordy A.P., Integrable equations in (2+1)-dimensions associated with symmetric and homogeneous spaces, J. Math. Phys. 28 (1987) 2018-2024 Sydorenko Yu.M. Binary transformations and (2 + 1)-dimensional integrable systems // Ukr. Math. journ. –

• 2002. – V. 54, №11. – p. 1859-1884.

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Integro-differential Lax representations for Davey-Stewartson and Chen-Lee-Liu equations Consider the following pair of operators: L1 = ∂y − qM0D−1r⊤D, (50) M2 = α2∂t2 −c1D2 −c2∂2

y +2c1S1D +2c2qM0D−1∂yr⊤D, (51)

where q = q(x, y, t2), r = r(x, y, t2) and S1 = S1(x, y, t2) are matrix functions with dimensions (N × M) and (N × N) respectively; M0 is a constant (M × M)-dimensional matrix. Equation [L1, M2] = 0 is equivalent to the following system:        α2qt2 − c1qxx − c2qyy + 2c1S1qx − 2c2qM0S2+ +2c2qM0(r⊤q)y = 0, α2r⊤

t2 + c1r⊤ xx + c2r⊤ yy + 2c1r⊤ x S1 + 2c2S2M0r⊤ = 0,

S1y = (qM0r⊤)x + [qM0r⊤, S1], S2x = (r⊤

x q)y.

(52)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Integro-differential Lax representations for Davey-Stewartson and Chen-Lee-Liu equations a). Under additional conditions α2 ∈ iR, c1, c2 ∈ R, M0 = −M∗

0, r⊤ = q∗, S1 = S∗ 1 operators L1 (50) and M2 (51)

are D-skew-Hermitian (L∗

1 = −DL1D−1) and D-Hermitian

(M∗

2 = DM2D−1). System (52) has a form:

α2qt2 − c1qxx − c2qyy + 2c1S1qx + 2c2qM0S2 = 0, S1y = (qM0q∗)x + [qM0q∗, S1], S2x = (q∗qx)y. (53) Consider a scalar case of equation (53) (N = 1, M = 1) and take c2 = 0, y = x, µ := M0. Then we obtain Chen-Lee-Liu equation (DNLS-II) from (53): α2qt2 − c1qxx + 2c1µ|q|2qx = 0. (54)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Integro-differential Lax representations for Davey-Stewartson and Chen-Lee-Liu equations References H.H.Chen, Y.C.Lee, C.S.Liu. Integrability of nonlinear Hamiltonian systems by inverse scattering method. Physica

• Scr. 20 (1979) 490–492.

D.J.Kaup, A.C.Newell. An exact solution for a derivative nonlinear Schrodinger equation. J.Math. Phys. 19:4 (1978) 798–801.

• V. S. Gerdjikov, M. I. Ivanov. The quadratic bundle of

general form and the nonlinear evolution equations. II. Hierarchies of Hamiltonian structures, Bulgarian J. Phys., v.10, N.2, 130-143, (1983)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Integro-differential Lax representations for Davey-Stewartson and Chen-Lee-Liu equations b). We will put M0r⊤(x, y, t2) = ν, where ν is (M × N)-dimensional constant matrix. After the change u := qν system (52) takes the form: α2ut2 − c1uxx − c2uyy + 2c1S1ux + 2c2uuy = 0, S1y = ux + [u, S1]. (55) System (55) is (2+1)-dimensional matrix generalization of Burgers equation. It can be generalized onto (n + 1)-dimensional case:

• α2ut2 = ∆u − 2S∇u,

∂Si ∂x1 = ∂u ∂xi + [u, S1], i = 1, n,

(56) where S = (S1, . . . Sn), ∆ = n

i=1 ∂2 ∂x2

i , ∇ = ( ∂

∂x1 , . . . , ∂ ∂xn ).

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Integro-differential Lax representations for Davey-Stewartson and Chen-Lee-Liu equations Proposition Let T := T(x, y, t2) be ((N × N))-matrix function that satisfies equation: α2Tt2 = c1Txx + c2Tyy (57) Then (N × N)-matrix functions u := −T −1Ty, S1 = −T −1Tx. (58) satisfy system (55).

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Integro-differential Lax representations for Davey-Stewartson and Chen-Lee-Liu equations Remark It can be checked that functions u, S1 defined by formula (58) satisfy another version of (2+1)-dimensional generalization of matrix Burgers equation:

• α2ut2 − c1uxx − c2uyy + 2c1S1ux + 2c2uuy = 0,

α2S1t2 − c1S1xx − c2S1yy + 2c1S1S1x + 2c2uS1y = 0 (59)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

It is also constructed the integro-differential representation for the equation: α3qt3 + c1qxxx − c2qyyy − 3c1µqx

• |q|2

xdy+

3c2µqy

• |q|2

ydx + 3c2µq

• (¯

qqy)ydx − 3c1µq

• (qxq)xdy = 0.

(60) where α3, µ, c1, c2 ∈ R, which can be reduced to the mKdV equation: α3qt3 + qxxx − 6µq2qx = 0. (61)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Lax integro-differential representation was also constructed for the following system: α3qt3+c1qxxx−c2qyyy−3c1v1qxx−3c1v3qx+3µc2qyD−1{¯ qqx}y+ +3c2µqD−1{¯ qqxy}y − 3c2µ2qD−1{|q|2¯ qqx}y = 0, v1y = µ(|q|2)x, v3y = µ(qx ¯ q)x − 2µv1(|q|2)x, (62) where α3, c1, c2 ∈ R, µ ∈ iR, v1 = v∗

1, v3 + v∗ 3 = v1x, which

reduces to the higher Chen-Lee-Liu equation (c1 = 1, c2 = 0): α3qt3 + qxxx − 3µ|q|2qxx − 3µ¯ qq2

x + 3µ2|q|4qx = 0.

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi

Thank you for your attention!